A Simple XY Model for Cascade Transfer

TL;DR

This paper introduces a modified XY model where local interactions produce cascade transfer, revealing an inverse energy cascade with a distinct $k^{-3}$ spectrum, differing from classical turbulence.

Contribution

The paper presents a new XY model demonstrating cascade transfer from local interactions, with theoretical and numerical analysis of energy flux and correlation functions in multiple dimensions.

Findings

Inverse energy cascade observed in the model

Energy spectrum proportional to $k^{-3}$

Distinct universality class from standard turbulence

Abstract

We propose a modified XY model in which cascade transfer emerges from spatially local interactions, where the spin corresponds to the "velocity" of a turbulent field. For this model, we theoretically calculate the scale-to-scale energy flux and the equal time correlation function in $d$ dimensions. The result indicates an inverse energy cascade with the non-Kolmogorov energy spectrum proportional to $k^{-3}$. We also numerically confirm the result for the cases $d=2$ and $d=3$. We thus conclude that the cascade transfer in our model represents a different universality class from standard fluid turbulence.